A Recipe for Constructing Frustration-Free Hamiltonians with Gauge and Matter Fields in One and Two Dimensions

Abstract

State sum constructions, such as Kuperberg’s algorithm, give partition functions of physical systems, like lattice gauge theories, in various dimensions by associating local tensors or weights, to different parts of a closed triangulated manifold. Here we extend this construction by including matter fields to build partition functions in both two and three space-time dimensions. The matter fields introduces new weights to the vertices and they correspond to Potts spin configurations described by an A-module with an inner product. Performing this construction on a triangulated manifold with a boundary we obtain the transfer matrices which are decomposed into a product of local operators acting on vertices, links and plaquettes. The vertex and plaquette operators are similar to the ones appearing in the quantum double models (QDM) of Kitaev. The link operator couples the gauge and the matter fields, and it reduces to the usual interaction terms in known models such as Z2 gauge theory with matter fields. The transfer matrices lead to Hamiltonians that are frustration-free and are exactly solvable. According to the choice of the initial input, that of the gauge group and a matter module, we obtain interesting models which have a new kind of ground state degeneracy that depends on the number of equivalence classes in the matter module under gauge action. Some of the models have confined flux excitations in the bulk which become deconfined at the surface. These edge modes are protected by an energy gap provided by the link operator. These properties also appear in “confined Walker-Wang” models which are 3D models having interesting surface states. Apart from the gauge excitations there are also excitations in the matter sector which are immobile and can be thought of as defects like in the Ising model. We only consider bosonic matter fields in this paper.

Lattice systems with gauge and matter fields is a recurring theme in high energy and condensed matter physics. They are a useful way of regularizing field theories in particle physics where the standout example is lattice QCD, a widely studied field [1]. The gauge groups involved in these programs are compact, continuous non-Abelian groups like SU(3) and SU(2) and Abelian groups like U(1). Whereas in the context of condensed matter physics such lattice systems are used to model magnetism in solids and crystals and study various phases of strongly correlated electrons most of which are typically low energy phenomena. One of the prominent examples is the Hubbard model which is an interacting microscopic model used to describe a variety of strongly correlated phenomena [2]. In these examples the gauge groups considered are finite discrete groups such as the Abelian Zn. The classic examples using such Abelian groups as degrees of freedom are the Ising chain, Heisenberg model and the Potts model [3]. An important connection between lattice spin systems and gauge theories was given in the famous article by Kogut [4, 5]. They study in detail the connection between a d-dimensional quantum spin system and the d+1-classical system via the transfer matrix method. This gives the correspondence between the two-dimensional quantum Ising model and three dimensional Z2 lattice gauge theory.

The partition functions of such lattice systems can be found out in a systematic way by looking at them as state sum models. These state sum models are used in several areas of mathematics and physics, including statistical mechanics [6], random matrices [7], knot theory [8], lattice gauge theory [9] and quantum gravity [10]. Such state sum models are defined based on a combinatorial decomposition of a manifold such as a lattice or a triangulation, which can be interpreted to be the space-time in a physical picture. In these models the degrees of freedom live on the vertices and/or edges of the lattice. In a more general setting one could associate local states on even faces and volumes. We restrict our attention to those living only on the vertices and edges. A state in such a system is the tensor product of the configuration in each local edge and vertex. Local weights are associated to the vertices edges and faces of the triangulated lattice. These weights can be thought of as tensors with indices which can be raised and lowered. These tensors are then allowed to contract and thus one obtains a number corresponding to the partition function when this construction is carried out for a closed triangulated manifold.

These constructions have been carried out in the past for finding 3-manifold invariants which were also interpreted as the partition functions of certain physical models. They were especially useful in constructing the partition functions of Topological Quantum Field Theories (TQFTs). An important example is the 3D Dikgraaf-Witten invariant [11] which furnished all topological lattice gauge theories. Hamiltonian realizations of these invariants in two dimensions is given in [12]. Another important example is the Turaev-Viro type of TQFTs in 3D [13]. These are realized by the Levin-Wen models or the string-net Hamiltonians [14] in two dimensions. These are exactly solvable models made up of commuting projectors describing degrees of freedom belonging to a unitary fusion category located on the edges of the lattice. They realize anyons as low energy excitations and provide a general class of long-ranged entangled topologically ordered phases in two dimensions. The Turaev-Viro invariant are known to be equivalent to the Chain-Mail link invariant [15]. It was shown in [16] how one can obtain the string-net models from the Chain-Mail link invariant which is also a knot invariant.

In the spirit of such computations we showed in [17] that the partition function of Quantum Double Hamiltonians (QDM) of Kitaev [18, 19] can be obtained by a deformation of Kuperberg’s 3-manifold invariants [20]. The QDM Hamiltonians realize lattice gauge theories based on an involutory Hopf algebra A. Lattice gauge theories based on a group G is a special case of this as A=C(G) is an involutory Hopf algebra. When G=Z2 we obtain the toric code model. Thus the quantum double Hamiltonian realizes a representation of the quantum double of these involutory Hopf algebras. This is a general recipe to construct quasi triangular Hopf algebras from a given Hopf algebra. These quasi triangular structures are governed by a R-matrix which form a representation of the braid group in two dimensions and hence help us find anyonic solutions [21]. In [17] we embed such quantum double models in a bigger parameter space namely the full parameter space of lattice gauge theories in three dimensions. For particular choices of the parameters we obtain the solvable models which describe the same phase as the QDM phase of Kitaev. For more general parameters we go away from this phase in a manner similar to the effect of perturbations which can be thought of as magnetic fields in the simple cases of the group algebra. By studying it in a bigger parameter space new topological and quasi-topological phases were also shown to exist [17] for these lattice gauge theories. These phases were understood by analyzing the Hamiltonians derived from the transfer matrices of these lattice gauge theories. These were constructed by carrying out Kuperberg’s prescription on a triangulated manifold with boundary. These transfer matrices are a product of local operators acting on the vertices, plaquettes and links. The vertex and plaquette operators are precisely the ones appearing in the QDM Hamiltonian [18, 19]. The ones acting on the links are like the perturbations to the QDM Hamiltonian. The trace of these transfer matrices coincide with the Kuperberg 3-manifold invariant. In this regard [17] is a significant extension of Kuperberg’s constructions as it is not limited to just obtaining 3-manifold invariants. By also parametrizing the transfer matrix we go away from the topological limits and recover the topological invariants for special parameters which incidentally also give the topologically ordered models.

The QDM and the string-net models are all examples of Hamiltonian realizations of lattice gauge theories based on involutory Hopf algebras and quantum groupoids or weak Hopf algebras [22] respectively. The gauge fields in these examples live on the edges of the three dimensional lattice. These are examples of long-ranged entangled (LRE) phases which form a large chunk of known topologically ordered phases [23]. The “other” chunk of known topological phases are the short-ranged entangled ones. These include the Symmetry Protected Topological (SPT) phases which are interacting bosonic phases which have edge states protected by a global symmetry group [24]. Conventional topological insulators [25] are non-interacting fermionic versions of these phases. Such phases are known to exist in all dimensions, however the physically interesting ones are the ones in one, two and three dimensions. Exactly solvable effective models describing such phases have been studied in [26, 27]. Gauging the global symmetries of these systems lead to fractional excitations in the bulk. These help mimic the behavior of fractional quantum Hall (FQH) states [28].

Apart from these two broad classifications of topological phases, the LRE phases were further subdivided when they are decorated with an additional symmetry giving rise to symmetry enriched topological (SET) phases [29, 30, 31]. These phases were studied in the mathematical framework of G-crossed braided fusion categories [32]. Exactly solvable lattice models for realizing SET phases with global symmetries were done in [33, 34, 35]. Gauging these global symmetries leads to new topologically ordered phases starting from the parent topological phase. Such phases were obtained locally in exactly solved Hamiltonians [?, 37, 38]. They were realized in more realistic systems, such as bilayer FQH states, by introducing dislocations and the accompanying branch cuts [39].

A common feature of the above systems is the existence of a global symmetry. In the case of the SPT phases these global symmetries are realized by using matter fields living on the vertices with no link degrees of freedom [24] and for the SET phases the global symmetry is realized by placing the system on an enlarged lattice [33, 34, 35] or by introducing dislocations in the lattice [39]. Thus it is desirable to go beyond the pure gauge systems in [17] by implementing an additional global symmetry which may help one obtain the SET and SPT phases in a systematic manner. We do this by extending the construction in [17] by including matter fields on the vertices and construct the corresponding transfer matrices of systems with gauge and matter fields. By using this transfer matrix we obtain exactly solvable models of such systems which also includes the interaction between the gauge and matter fields. These Hamiltonians are frustration free [40] and possess both a global symmetry, in the matter sector and a local symmetry in the gauge sector. They are frustration free in the sense that the Hamiltonian is a sum of terms such that the ground states of the full Hamiltonian are the lowest energy states of each individual term. The global ground states are also local ground states and hence there is no frustration or energy increase when all the terms in the Hamiltonian are included. A Hamiltonian which is a sum of commuting projectors satisfies these conditions and we shall see that the Hamiltonians constructed in this approach are of this form.

Before we provide a description of the construction of these transfer matrices we briefly write down some properties of the systems obtained from these transfer matrices. As noted before the Hamiltonians are exactly solvable and the spectrum is gapped. The main input in building these systems is the gauge group and the matter module which is acted upon by the gauge group and the choice of the representation of the gauge group on the matter module. Depending on this choice we have can obtain a variety of systems. A common feature to all of them is the ground state degeneracy which is no longer a topological invariant like in the pure gauge case [17] but rather develops another kind of degeneracy which depends on the number of equivalence classes of the matter module under the gauge action. Depending on the choice of the action of the gauge group the system can also develop a topological degeneracy in addition to the one coming from the different equivalence classes. These systems have excitations in both the gauge and the matter sectors. Unlike the deconfined excitations of the LRE phases, only the charge excitations are deconfined in this model. Depending on the choice of the action of the gauge fields on the matter degrees of freedom the system may or may not have deconfined fluxes. When the fluxes are confined, they are done so by string tension terms provided by operators acting on links as we shall see soon. However when these systems are placed on a manifold with a boundary the fluxes get deconfined and become edge modes which are protected by an energy gap provided by the link operators. This feature will be explained with a specific example in the text. This property of confined bulk excitations and deconfined excitations on the edges has been observed in confined Walker-Wang models [41] elaborated in [42, 43, 44]. The confined Walker-Wang models are systems in 3D, which when placed on manifolds with boundaries result in topologically ordered surface states. The examples in this paper achieve this in two dimensions through gapped edge modes.

We briefly describe our method to construct the transfer matrix of a lattice system with gauge and matter fields in two and three dimensions thus corresponding to quantum lattice models in one and two dimensions respectively. For convenience we only work with the case of finite groups though our methods can be extended to the case of continuous compact groups and in general to involutory Hopf algebras. Thus our focus is only on models which are pertinent for condensed matter systems. In particular we will be interested in finding exactly solvable Hamiltonians which act as effective descriptions of interacting systems of gauge and matter fields in one and two dimensions. We will explain the procedure for the three dimensional case. The procedure can be easily followed in the two dimensional case as well and is included as an appendix to this paper.

The construction proceeds as follows. The data needed to define a partition function is a triangulated, closed 3-manifold which we take to be of the form Σ×S1, where Σ is the two dimensional spatial slice, an involutory Hopf algebra A which make up the gauge degrees of freedom located on the edges of the triangulated 3-manifold and a n-dimensional vectorial space Hn, which carries a representation of A5 and has a co-structure and an inner product. The elements of Hn make up the matter fields which are located on the vertices of the triangulated 3-manifold. The invariant is built by contracting local weights associated to vertices, edges and faces. These local weights are built out of the structure constants of the involutory Hopf algebra A and the co-algebra Hn. These structure constants from A are given by tensors mcab, Δabc and Sba. They correspond respectively to the multiplication (product) map, coproduct map and the antipode. The structure constants from Hn are given by tensors tαβγ and μαβa where the former is similar to the coproduct for coalgebras and it arises from the costructure of Hn and the latter is the map which says how the gauge fields act on the matter fields, that is it is the map which furnishes a representation of A on Hn. In the structure constants listed the Latin alphabets index the gauge degrees of freedom and the Greek alphabets index the matter degrees of freedom. The scalar constructed out of these structure constants can be thought of as the result of contracting a three dimensional tensor network where we now consider the local weights as local tensors associated to the various parts of the three dimensional lattice. The partition function Z is parameterized by z and z∗, elements of the centers of the algebra A and its dual, mV an element of Hn and G the inner product in Hn. Thus we obtain Z(A,Hn,z,z∗,mV,G) which is in general a scalar and need not be topologically invariant.

Obtaining an operator from this scalar is a natural step when we carry out the above procedure on a 3-manifold with boundary. This operator is precisely the transfer matrix as it’s trace is the partition function. The 3-manifold we consider is Σ×[0,1] where as before Σ is the two dimensional spatial slice and the unit interval is along the third direction which we can think of as the Euclidean time direction. Thinking of the partition function as a contracted tensor network we now have non-contracted indices on the spatial slices Σ×{0} and Σ×{1}. This results in the transfer matrix U for the lattice system. However since we distinguish spatial and “time” directions in this construction we have more parameters for U. This means that the weights associated to the two directions are not necessarily the same. Denoting the space and time directions by S and T respectively we now have the fully parametrized transfer matrix as U(A,Hn,zS,zT,z∗S,z∗T,mV,GS,GT). We show how this global operator can be written as a product of local operators acting on vertices, links and plaquettes. We obtain

U(zS,zT,z∗S,z∗T,mV,GS,GT)

=

∏pBp(zS)∏l~Cl(GS)∏l(Tl(z∗S)DlLl(zT))

(1)

∏v~Vv(GT)Qv(mv)∏vAv(z∗T)

where v, l and p denote the vertices, links and plaquettes respectively.

This operator is very general and encompasses a wide variety of interacting lattice models with gauge and matter fields. However for specific values of the parameters, namely

zT

=

η,

z∗S

=

Aϵ,

mV

=

Hnϵ,

(GT)αβ

=

δ(α,β),

where η and Aϵ are the unit and co-unit of the algebra A and Hnϵ is the counit of Hn, we obtain exactly solvable models. The set of parameters we will work with in this paper are zS, z∗T and GS which give us

U(zS,z∗T,GS)=∏pBp(zS)∏l~Cl(GS)∏vAv(z∗T).

(2)

These give us models which are similar to the toric code in the sense of a frustration-free Hamiltonian made up of commuting projectors and having long ranged entangled ground states, but including matter fields. This Hamiltonian is given by

H=−∑vAv−∑pBp−∑lCl

(3)

where the vertex operators are gauge transformations just as in the QDM case, the plaquette operators measure gauge fluxes around a plaquette, and the link operator which is a gauged term with Potts-like nearest neighbor interactions. In other words, Cl couples the gauge configurations at the links with the Potts spins located at the vertices.

The contents of this paper are organized as follows. In section 2 we describe the construction of the transfer matrix in 2+1 dimensions in a systematic manner starting from the method to construct the weights/local tensors using the structure constants of the input algebra for the gauge algebra and its corresponding vector space carrying it’s representation. The partition function resulting from this assignment is briefly sketched before the transfer matrix is obtained from this using the splitting procedure. The splitting procedure leads to the transfer matrix written as a product of local operators. All these local operators are obtained at the end of section 3 as tensor networks, written down in the Kuperberg notation [20]. The algebraic expressions for these operators are given for the gauge algebra being the group algebra and the vector spaces being the ones carrying their representations, in section 4. The input algebra can be more general than group algebras with group algebras being most relevant for condensed matter physics. Some examples of exactly solvable models obtained from this construction are presented in section 5. A brief outlook makes up section 6. There are two appendices intended as a supplement to the main text. Appendix A furnishes the details of the basic input for our construction. In appendix B we show the same construction in 1+1 dimensions which produces quantum models in 1D. These can be used to obtain spin chain models and hence completes our formalism for physically relevant instances. Finally in appendix C we look at two examples of exactly solvable models in one dimension and find them to have interesting properties which warrant further study.

In order to define our model we define a lattice L which is a discretization of some (2+1)−manifold Σ×S1, where Σ is some compact 2−manifold such that ∂Σ=∅. For the purpose of this work it is enough to consider L a square lattice as the one shown in figure a. This lattice is constructed by gluing vertices, links and faces as shown in figure b.

(a) A piece of a (2+1)D square lattice.

(b) Small pieces of which the (2+1) lattice is made of.

Figure 1: Square lattice and its pieces.

The model we build has degrees of freedom associated to gauge fields (living on the links) and degrees of freedom associated to matter (living on the vertices). They are quite general in the sense that it can support models such as the quantum double models which includes the toric code, besides models in other phases of matter. The Hamiltonian operator is the one which propagates the states along the time direction and it is made up of a set of projectors operators which act on specific parts of the lattice. In this paper we will show how to get the Hamiltonian operator in the language of tensor networks. The way we proceed is very similar to the way we have done so in [17]. We start with a partition function Z written in the formalism of the state sum model which leads to a one step evolution operator U such that 6

Z(zS,zT,z∗S,z∗T,mV,GS,GT)=tr(UN),

where zS,zT,z∗S,z∗T are elements of the center of algebra and co-algebra which play role of parameters of the model, the subscript S and T means spacelike and timelike parameters while N is the number of steps in the time evolution. From this transfer matrix we shall be able to get a Hamiltonian H by taking its logarithm, i.e.,

U=U(zS,zT,z∗S,z∗T,mV,GS,GT)=e−ΔtH.

In the next section we will define the function partition function Z(zS,zT,z∗S,z∗T,mV,GS,GT) and relate it with a one step evolution operator. The partition function is made of a bunch of tensors associated to the vertex, links and faces of the lattice, all of them contracted with each other as we will see next.

2.1 The Partition Function

The partition function of the model is defined on an oriented square lattice L. We choose a square lattice for convenience. It can be easily defined on an arbitrarily triangulated lattice. The lattice is made up of vertices, links and faces glued together as shown in figure b. A tensor is associated for each vertex, link and face of L and contracted according to the gluing rules described below. In the next section we will show how the tensors we use to define the partition function are related with the structures constants of A and Hn.

Associating Tensors to the Lattice

The procedure outlined helps define a function L→C.

(a) The tensor Ma1a2a3a4 associated to a plaquette of the lattice.

(b) The tensor Tα1α2α3α4α5α6 associated to a vertex of the lattice.

(c) The tensor Labcdαβ associated to a link of the lattice.

Figure 2: The indices labelled by Latin letters (black arrows in the Kuperberg diagram) stand for the gauge fields while the indices labelled by Greek letters (green arrows in the Kuperberg diagram) stand for matter fields.

A tensor Ma1a2a3a4 is associated to each face. The four indices label the four links glued to this face. This is shown in figure a. The indices are ordered counter clockwise according to orientation of the plaquette. Each vertex is glued into six different links and hence the tensor associated to it is denoted Tα1α2α3α4α5α6 as shown in figure b. Finally each link is glued into four faces and two vertices, and so the tensor associated to a link is Labcdαβ as shown in figure c. Note that the indices labelled by Latin letters (black arrows in the Kuperberg diagram) stand for the gauge fields while the indices labelled by Greek letters (green arrows in the Kuperberg diagram) stand for matter fields. In figure 2 the dotted lines mean that something is going to be glued to it.

Contraction Rules

To take care of the orientation of the lattice we still need two more tensors which will play the role of fixing orientation of the gauge part and the matter part. The one associated to the gauge part is the antipode map S of the Hopf algebra A while the one associated to the matter part is some bilinear map G:Hn⊗Hn→C which we will describe later. The Kuperberg diagram for them are the ones shown in figure 3.

(a) The orientation tensor associated to the gauge part.

(b) The orientation tensor associated to the matter part.

Figure 3: The tensors which take into account the orientation of the lattice.

The contraction tensors with the contraction rules are shown in figure 4. For the gauge part (contraction of a plaquette with a link) we use the antipode tensor when the orientation does not match or we contract the tensors Ma1a2a3a4 with Labcdαβ directly as shown in figure a. For the matter part we contract the green arrow which is going out of the tensor Labcdαβ using the tensor Gαβ while the green arrow coming in the tensor Labcdαβ we contract directly, as shown in figure b

(a) The contraction rule for the tensors in the gauge sector.

(b) The contraction rule for the tensors in the matter sector.

Figure 4: Contraction Rules.

The partition function is defined as the contraction of all these tensors associated to the vertices, links and faces. Since the lattice we are considering has no boundary, ∂L=∅, there will be no free indices remaining after the gluing resulting in a scalar. As we will see in the next section the tensors Ma1a2a3a4 and Labcdαβ depend on the structure constants of A, Hn, elements of the center of the algebra A and its dual, elements of Hn. With these parameters the partition function can be written as

where the products runs over the orientations (o), the plaquettes (p), links (l) and vertices (v) of L.

2.2 The Transfer Matrix U from the Partition Function Z

Having defined the partition function we can now define a one step evolution operator U such that

Z=tr(UN).

The way to do this is by looking at the lattice in which the partition function is defined on and to take slices of it in the time direction. We now introduce a new point of view where we look at the lattice as a contraction of a bunch of tensors as defined above. As we will see sometimes it will be convenient to think of the lattice as a tensor network and sometimes more convenient to think of it as a gluing of vertices, links and faces.

Figure 5: The partition function as the trace of a product of a one step evolution operator.

In figure 5 the black and green arrows represent the free indices (gauge and matter degrees of freedom) of the one step evolution operator. The one step evolution operator is the one shown in figure 6, it is made of a bunch of boxes without their caps glued one besides the other.

We will see that the one step evolution defined in figure 6 is a product of local operators. These local operators act on vertices, links and plaquettes of the lattice, and hence are called vertex, link and plaquette operators, respectively. But before we split it we must have a better definition of the tensors used to build it. In other words we need to know how they are related with the structure constants of A and Hn. In the next section we will define their algebraic structure which is a very important part of the model.

3.1 The Structure Constants

Let A be an involutory Hopf algebra with a product m:A⊗A→A, a co-product δ:A→A⊗A and a involutory map S:A→A. Its structure constants are the tensors shown in figure 7.

(a) The multiplication map.

(b) The co-product map.

(c) The antipode map.

Figure 7: Structure constants of the involutory Hopf algebra A.

Let Hn be a module of A which has a semi-simple co-algebra structure t:Hn→Hn⊗Hn. We define the action of A on Hn by the map μ:A⊗Hn→Hn in the following way

μ:a⊗v↦μ(a)⊳v.

The structure constants involving Hn are the ones shown in figure a and b. We also show the action of the algebra A on Hn with respect to the product of the algebra A, in other words we want this action to be a homomorphism, i. e., μ(ab)⊳v=μ(a)⊳(μ(b)⊳v)∀v as shown in figure c.

(a) The action map of A on Hn.

(b) The co-structure map in Hn.

(c) The algebra action on the module is a homomorphism.

Figure 8: Structure constants involving the Hn space.

3.2 Building the Tensors

The first tensor we will define is the tensor Ma1a2a3a4. Its definition is shown in figure 9. Due to the associativity of the algebra the tensor Ma1a2a3a4 is invariant under cyclic permutation of its indices and can be written in different ways as shown in figure 9. The element z is an element of the center of the algebra.

Figure 9: The definition of the tensor Ma1a2a3a4.

The tensor Tα1α2α3α4α5α6 is made of the co-structure map in Hn as shown in figure 10. As the co-product is co-commutative this tensor is invariant under interchange of a pair of its indices. Like the tensor Ma1a2a3a4 this tensor can also be written in different equivalent forms as shown in figure 10.

Figure 10: The definition of the tensor Tα1α2α3α4α5α6.

Finally the tensor associated to the links given by Labcdαβ is defined in figure 11. As we can see it is made up of the algebra action on Hn given by the μ map and the tensor Δa1a2a3a4a5 which is also defined in figure 11. The element z∗ is an element of the co-center of the algebra A∗.

Figure 11: The definition of the tensor Labcdαβ.

3.3 Splitting the One Step Evolution Operator

In order to make the procedure clear we will represent the spacelike part as being made of spacelike plaquettes and spacelike links and the timelike part as being made of timelike plaquettes and timelike links. The splitting will be done in two steps: first we will split the tensors associated to spacelike links and vertices and then we split the tensor associated to the timelike plaquettes. These two steps show how the one step evolution operator can be written as a product of local operators. Although the model does not depend on the orientation of the lattice we have to set up some orientation in order to use the contraction rules described above. So we consider the orientation shown in figure 12.

Figure 12: The orientation of the lattice considered.

Splitting the Spacelike Part of the Tensor Network

Consider a spacelike link which is shared by two adjacent plaquettes and two vertices as shown in figure 13.

Figure 13: A spacelike link shared by two adjacent plaquettes and two vertices.

The tensor network associated to the figure 13 is shown in figure 14. In the picture tlp and tll stand for timelike plaquette and timelike link respectively.

As seen in figure 11 the tensor Labcdαβ can be splitted in terms of the structure constants representing the co-product of the algebra. Thus the diagram in figure 14 reduces to the one shown in figure 15.

Figure 15: Tensor network associated to the picture in figure 13 with the tensor Labcdαβ splitted.

To make things clear we write down the same tensor network of figure 15 in figure 16, where we have separated the tensors associated to each one of the spacelike plaquettes. Each leg of the tensor Ma1a2a3a4 will be contracted to a tensor δcda (directly or indirectly by the antipode map) and that will lead to the plaquette operator as we shall soon see.

We now split the tensor associated to each vertex in figure 16. For that we just use the definition of the tensor Tα1α2α3α4α5α6 in figure 10. The splitting shown in figure 17 has to be done for the all the vertices of the lattice, but here for simplicity we are illustrating just for one single vertex.

Figure 17: Splitting the tensors associated to a vertex.

The tensor network in figure 16 is rewritten in a separated way as shown in figure 17. After all this splitting we see that the spacelike tensor network part can be written as a product of two blocks of tensor networks which are the ones shown in figure 18. The first one (figure a is the operator called the plaquette operator. The second one in figure c is called Kl and it will be contracted with the timelike tensor network part to build the others operators which make up the transfer matrix.

(a) The plaquette operator as a tensor network.

(b) The plaquette p where the operator Bp acts on.

(c) The tensor network which will be contract with the timelike part of the remaining tensor network.

Figure 18: The spacelike part of the tensor network can be written as a product of these two blocks of tensor networks.

Splitting the Timelike Part of the Tensor Network

We now split the timelike part of the tensor network. For that consider a timelike plaquette which is shared by two timelike links and one spacelike link, as shown in the figure a. The spacelike link on the top is the one which has already been splitted and it led to the tensor network attached to the timelike plaquette as shown in figure b.

(a) A timelike plaquette of the transfer matrix.

(b) A timelike plaquette with the tensor network associated to the spacelike link attached above.

Figure 19: Timelike part of the transfer matrix.

The tensor network associated to the timelike plaquette contracted with the two timelike links is the one one shown in figure a and it can be easily changed to the one in figure b by using the definition of the tensor Ma1a2a3a4 in figure 9.

(a) The tensor network associated to the timelike part of the diagram.

(b) The tensor Ma1a2a3a4 splitted in terms of the structure constants of the algebra.

Figure 20: Splitting the tensor network of the timelike part of the transfer matrix.

As before we rewrite the tensor network in figure b as the one in figure 21.

Figure 21: Rewriting the tensor network in figure b with the tensors separated.

In order to keep it clear in our mind where are these tensors acting on one should take a look at the figure b. Note that each and every timelike tensor Labcdαβ will have a tensor mcab contract with each of its gauge legs (directly or indirectly with the antipode). It lead to an operator Av for each timelike link of the lattice, or in other words, for each vertex spacelike vertex v. This operator is the one drawn in figure a. The tensor in highlight in figure 21 is now attached to the tensor network on the top in figure b and it give us a new operator Kl which acts on the links, see figure c

(a) The vertex operator.

(b) The vertex v where the vertex operator Av acts.

(c) The tensor network of the link operator.

Figure 22: Vertex operator and the operator Kl which acts on a link.

The operator Kl is the same on which appears in the (1+1) dimensional models. In the appendix we show how the transfer matrices of such a models can be obtained e also that the Kl operator can be written in terms of simplers operators. The transfer matrix we started with can now be written as

U(zS,zT,z∗S,z∗T,mV,GS,GT)=∏pBp(zS)∏lKl∏vAv(z∗T).

But the Kl operator, as can be seen in the appendix, can also be written as

∏lKl=∏lCl(GS)∏l(Tl(z∗S)DlLl(zT))∏v~Vv(GT)Qv(mv),

where the operators Qv(mv), Tl(z∗S) and Ll(zT) will be explained in the next section. The operators Dl and ~Vv(GT) are explained in the appendix. The operator Cl(GS) is called the link operator and acts on a link and on the its vertex. This operator is defined below in figure a.

So far we have used a diagrammatic language where we associated tensors to the vertices, links and plaquettes of a triangulated three manifold to build a tensor network and showed that this results in a partition function for a closed manifold and a transfer matrix for a manifold with boundaries. We then used this transfer matrix represented by a tensor network to find out the local operators which can be used to piece up such a network. In this section we will write down the algebraic expressions for these operators and study their properties. For simplicity we choose the gauge algebra to belong to groups and so the states on the links are elements of the group algebra of a gauge group denoted by C(G). The matter degrees of freedom belong to a module of C(G) which we denote by Hn, in other words Hn is a vector space carrying an action of C(G). The formalism developed so far can be applied to any involutory C∗-Hopf algebra and its corresponding module.

The operators that make up the transfer matrix can be divided into the ones which act only on the gauge fields, the ones that only act on the matter fields and the ones that involve the coupling between the gauge and the matter fields through the μ map defined in figure (11). We describe each set separately before we write down the full transfer matrix.

In order to write an algebraic expression for the operators we derived in previous section we need to define three operators which are made of the structure constants of the algebra and the module, as shown in figure 24.

(a) left multiplication.

(b) right multiplication.

(c) gauge projector.

(d) matter projector.

Figure 24: These operators are the ones which we use to build the vertex, plaquette and link operators.

These operators are linear on its parameters, in other words, L(z)=∑gzgL(ϕg) (the same for R(z), T(z∗) and Q(v)) and they act on the vector basis as

L(ϕg)|h⟩

=

|gh⟩,

R(ϕg)|h⟩

=

|hg⟩,

T(Ψg)|h⟩

=

δ(g,h)|h⟩,

Q(χi)|j⟩

=

δ(i,j)|j⟩.

(4)

where we have used the bra-ket notation for elements of the basis (|g⟩:=ϕg and |i⟩=χi). Sometimes we use the short notation Lg=L(ϕg), Rg=R(ϕg), Tg=T(Ψg) and Qi=Q(χi). Now using this operators in the next section we write the plaquette, vertex and link operators derived before in terms of them. There are three operators acting on the gauge sector alone. These are given by the plaquette operator and two operators acting on the qudits on the links. They can be deduced from their respective tensor network representation from the previous sections. We write down each of these in what follows. We will also see how the different parameters parametrizing the transfer matrix U(zS,zT,z∗S,z∗T,mV,GS,GT) arise in the definition of these operators. This will also show how the transfer matrix depends on these parameters.

4.1 The Gauge Sector

The plaquette operator can be written down by looking at its tensor network representation given in figure (a). We find

Bp=∑C∈[G]βCBCp

(5)

where C labels a particular conjugacy class from the set of conjugacy classes in G denoted by [G]. Each of BCp is given by

BCp=|G|∑g∈C∑{gi}δ(g1g2g3g4g,1G)Tg−11i1⊗Tg2i2⊗Tg3i3⊗Tg−14i4

(6)

where 1G is the identity of the group G.

While writing down the plaquette operator in Eq.(5) we have made use of the fact that the tensor associated to the spacelike plaquette Ma1a2a3a4 contains a central element of C(G) as can also be seen in figure (9). For the plaquette operator in Eq.(5) this central element zS, where S denotes spacelike, is given by

zS=∑C∈[G]βC∑g∈Cϕg

(7)

with the set {ϕg} forming a basis of C(G).The action of the plaquette operator on the oriented square lattice is shown in figure b.

The operators BCp form a complete basis of orthogonal projectors since

BCpBC′p=δ(C,C′)BCp,

and also

∑CBCp=I.

One of the link operators acting on the qudit on the links is given by

Ti(z∗S)=∑g∈GbgTgi

(8)

where bg are constants and Tgi is gauge projector defined in figure c. These operators arise by the parameter on spacelike links given by z∗S which is a central element of the dual of C(G). They can also be seen in figure (11). This central element is given by

z∗S=∑g∈Gbgψg

(9)

with the set {ψg} forming the basis of the dual of C(G).

The second link operator acting in the gauge sector is given by

Lj(zT)=∑R∈IRR's ofGaRLj(zR)

(10)

where aR’s are constants and zR is given by

zR=1|GR|∑g∈GχR(g)ϕg.

(11)

χR(g) is the character of the element g in the IRR R and |GR| is the number of elements with non-zero trace in the IRR R. The parameters zT is given by

zT=∑RaRzR

(12)

We could as well have defined the operator Rj(zT), but since zT is an element of the center of the algebra these two operators are the same, namely Lj(zT)=Rj(zT).

The operator Xj in Eq.(10) is obtained through the parameter zT in the timelike plaquette. This is an element of the center of C(G) given by

zT=∑RaR∑g∈GχR(g)ϕg.

(13)

These operators exhaust the operators acting only in the gauge sector. We now turn to those acting only in the matter sector.

4.2 The Matter Sector

There are two operators which act only in the matter sector. One of them is obtained by parametrizing the vertices by mV, an element of the module Hn and the other by parametrizing the timelike links with the inner product GT. The operator parametrized by mV is given by

Qv(mv)=n∑i=1ciQiv

(14)

where ci are constants and Qiv is given by

Qiv|j⟩=δi,j|j⟩

(15)

where χi,j are elements of the module Hn. For the definition of Qv in Eq.(14) the element mV is given by

mV=n∑i=1ciχi.

(16)

As the module only has a co-structure this operator is very similar to the coproduct map in the gauge sector. In fact for the module Hn with symmetry group Zn we can identify Qjv with the Tωj operator of the gauge sector for C(Zn) with ω=e2πin and j∈(0,⋯,n−1). If we label the elements of the group Zn as {ωk;k∈(0,⋯,n−1)} we can then write Qlv as

Qlv=1nn−1∑k=0χωl(ωk)Zkv

(17)

where χωl(ωk) is the character of the element ωk in the IRR of Zn labelled by ωl. There are n such expressions corresponding to the n IRR’s of Zn.

We can also consider modules Hn with other symmetry groups, especially non-Abelian groups, and in this case the expression for Qiv is different from the one given by Eq.(17). The symmetry groups of the module Hn can also be thought of as global symmetry groups of the system.

We now consider the operators acting on both the gauge and matter sectors.

4.3 The Gauge + Matter Sector

There are two operators acting on both the gauge and matter sector. They are the operators coupling the two sectors. These operators are the vertex and link operators given in figures (a) and (a) respectively.

The vertex operator Av is given by

Av=∑g∈GαgAgv

(19)

where

Agv=μv(ϕg)⊗Lgj1⊗Lgj2⊗Rg−1j3⊗Rg−1j4

(20)

with μv(ϕg) being the representation of the gauge group on the matter field located on the vertex v. The single qudit operators Lgi and Rgi, acting on the gauge fields located on the links, are given by Eq.(4).

The vertex operator Av is obtained by using the parameter z∗T in the transfer matrix given by

z∗T=∑g∈Gαgψg.

(21)

This is a parameter living on the timelike links of the 2+1 dimensional manifold.

For particular choices of the parameter z∗T we obtain the set of orthogonal vertex operators which are projectors. These parameters are given by

z∗T=1|GR|∑g∈GχR(g)ψg

(22)

for the different R’s in the set of IRR’s of G.
They result in the following set of vertex operators

ARv=1|GR|∑g∈GχR(g)Agv.

(23)

This vertex operator is very similar to the one defined for the quantum double models [17, 18, 19] with the addition of the representation of the gauge group acting on the vertex part. Thus it can still be thought of as a gauge transformation just as in the case of the quantum double models.

The other operator acting on both the gauge and the matter sector is the link operator represented as a tensor network in figure (a). In terms of operators this operator is given by

Cl=∑χv1,χv2,ϕlGS(μ(ϕl)χv1,χv2)Qχv1v1TϕllQχv2v2

(24)

where the matrix GS implements the inner product between the vectors in the module Hn.

This inner product G can also be thought of to be represented by the following tensor network shown in figure (25),

Figure 25: The tensor network representation for the inner product G in terms of the matrix R.

where T−1 is a tensor such that (T−1)αβTβγ=δ(α,γ), in other words, (T−1)αβ=δ(α,β).

The matrices R in this definition can be thought of as those performing basis transformations on the elements of the module Hn. If the matrix G is chosen to be identity, it is the same as choosing R to be identity. For other choices of R we can find the operators orthogonal to Cl. This will soon become clearer when we illustrate using examples.

We thus have the operators forming the transfer matrix of a lattice theory with gauge and matter fields. To sum up these operators include the vertex, plaquette and link operators along with the single qudit operators acting in the gauge sector given by Eq.(8) and Eq.(10) and in the matter sector given by Eq.(14). We now proceed to study the algebra between these operators.

4.4 Algebra of the Operators

The remaining basic operators in the theory are given by Lgi, Rgi, Tgi and Qiv. Clearly Tgi and Qiv are projectors and Qiv commutes with the remaining operators as they act on different sectors. We have the following relations

Lg1iLg2i

=

Lg1g2i

(25)

Rg1iRg2i

=

Rg2g1i

(26)

Tg2iLg1i

=

Lg1iTg−11g2i

(27)

Tg2iRg1i

=

Rg1iTg2g−11i.

(28)

Using these relations it is easy to see that the vertex and plaquette operators given by Eq.(19) and Eq.(5) commute. This computation is exactly similar to the computation used to prove the commutativity of these two operators in the case of the quantum double models. This is true because the plaquette operators in our case of theories with gauge and matter fields is left unchanged with respect to the case with pure gauge fields.

The link operators Cl trivially commute with the plaquette operators as both are diagonal. This can be seen from their respective definitions given by Eq.(24) and Eq.(5). The only thing that we have to prove is the commutation between the vertex operator and the link operator in their common region of support.

Consider the action of the operators Av and Cl on the gauge and matter degrees of freedom. The proof then goes as follows for their action on the common support, we distinguish two cases as the vertex operator action depends on the orientation of the lattice, the first case being the one when the vertex operator acts on the vertex at the left of the edge l, this is

Agv1Cl\Ket…,χv1,ϕl,χv2,…

=⟨μ(ϕl)χv1,χv2⟩GAgv1\Ket…,χv1,ϕl,χv2,…

=⟨μ(ϕl)χv1,χv2⟩G\Ket…,μ(ϕg)χv1,ϕlϕg−1,χv2,…,

(29)

whereas

ClAgv1\Ket…,χv1,ϕl,χv2,…

=Cl\Ket…,μ(ϕg)χv1,ϕlϕg−1,χv2,…

=⟨μ(ϕlϕg−1)μ(ϕg)χv1,χv2⟩G\Ket…,μ(ϕg)χv1,ϕlϕg−1,χv2,…,

(30)

since the module map is a group homomorphism we note that:

μ(ϕlϕg−1)μ(ϕg)

=μ(ϕl)μ(ϕg−1)μ(ϕg)

=μ(ϕl)μ(ϕg−1ϕg)

=μ(ϕl),

thus, the coefficient on the right hand side of eq.(4.4) now is the same as the one in eq.(4.4), hence [Agv1,Cl]=0. Let us now consider the case when the vertex operator is acting at the right of the edge l, i.e.

Agv2Cl\Ket…,χv1,ϕl,χv2,…

=⟨μ(ϕl)χv1,χv2⟩GAgv2\Ket…,χv1,ϕl,χv2,…

=⟨μ(ϕl)χv1,χv2⟩G\Ket…,χv1,ϕgϕl,μ(ϕg)χv2,…,

(31)

while

ClAgv2\Ket…,χv1,ϕl,χv2,…

=Cl\Ket…,χv1,ϕgϕl,μ(ϕg)χv2,…

=⟨μ(ϕgϕl)χv1,μ(ϕg)χv2⟩G\Ket…,χv1,ϕgϕl,μ(ϕg)χv2,…,

(32)

for the commutation to hold we require:

⟨μ(ϕg)χα,χβ⟩G=⟨χα,μ(ϕg)†χβ⟩G,

(33)

this means the representation map μ is unitary, equivalently

μ(ϕg)†=μ(ϕg−1).

(34)

therefore the right hand side of eq.(4.4) reduces to that of eq.(4.4). Thus, [Agv2,Cl]=0.
So, we just showed that [Agv,Cl]=0 for any ϕg∈C(G) and for any vertex v of the lattice, thus it follows that [Av,Cl]=0.

Moreover we only consider μ’s which permute the basis elements of the matter module. Under this condition ⟨μ(ϕl).χv1|χv2⟩ is real which makes the link operator Cl hermitian. This also ensures that the vertex operator is hermitian as the gauge part of the vertex operator is the same as the one in the QDM of Kitaev. The plaquette operator is the same as those appearing in the QDM of Kitaev. Thus all the operators in the Hamiltonian are hermitian making the evolution unitary.

4.5 The Transfer Matrix

We are now in a position to write down the full transfer matrix of a lattice theory with gauge and matter fields as a product of these local operators. Thus we have

U(zS,zT,z∗S,z∗T,mV,GS,GT)

=

∏pBp(zS)∏lCl(GS)∏l(Tl(z∗S)DlLl(zT))

∏v~Vv(GT)Qv(mv)Av(z∗T)

The other operators namely Bp(zS), Tl(z∗S), Qv(mv), Ll(zT), Av(z∗T) are given by Eq.(5), Eq.(8), Eq.(14), Eq.(10), Eq.(19) respectively. Through these operators the transfer matrix U obtains its dependence on zS, z∗S, mv, zT and z∗T respectively.

The operator ~Cl is defined as a linear combination of orthogonal projectors. We write it as

~Cl=γCl+γ⊥C⊥l

(35)

where Cl is given by Eq.(24) and C⊥l is the operator orthogonal to Cl. The form of the orthogonal operator C⊥l depends on the module Hn considered. In general there are several operators orthogonal to the operator Cl. For the modules with symmetry group as Zn we can write down a general form for these orthogonal operators. If we denote the inner product in the module Hn as ⟨χi|χj⟩ then we can write down the operator Cl as

Cl|χv1,ϕl,χv2⟩=⟨μ(ϕl).χv1|χv2⟩|χv1,ϕl,χv2⟩.

(36)

We can now write n−1 other orthogonal operators as

C⊥l|χv1,ϕl,χv2⟩=⟨μ(ϕl).χv1|Xi|χv2⟩|χv1,ϕl,χv2⟩;i∈{1,⋯,n−1}

(37)

where X is the shift operator generating Zn given by

Xk|χi⟩=|χi+k⟩.

(38)

The transfer matrix gets its dependence on G through the link operator Cl. The inner product given by G in turn depends on the matrix R as shown by it’s tensor network representation in figure (25).

Thus for each i in Eq.(37) we choose a different R matrix in the inner product G. For a given i in Eq.(37) R is given by (Xi)12. This matrix R can be thought of as a basis transformation in the module Hn.

Having written down the general expression for the transfer matrix U of a lattice theory with gauge and matter fields we will now consider specific examples which illustrate the formalism developed so far.

We will consider two examples of Hamiltonians having long-ranged entangled ground states derived from the construction we have illustrated. The first example is the simplest Hamiltonian with the gauge group being Z2 and the vector space carrying its representation being the two dimensional vector space H2. We will denote this model as H2/Z2. The second example is a slight modification with the vector space H2 replaced by H3, a three dimensional vector space carrying the representation of the gauge group Z2. We denote this model as H3/Z2. We consider each of them separately in what follows.

The ground state degeneracies in each of these cases is a topological invariant and can be computed numerically. There are no winding operators as in the toric code for the H2/Z2 case but we do have a winding operator for the H3/Z2 case. We will write down the ground states in both the cases and give their tensor network representations.

Both these examples have confined excitations apart from one deconfined excitation. We will briefly look at the vertex and gauge excitations in the H2/Z2 case. The vertex excitations of H3/Z2 were studied in detail in [45]. Here we will just look at it’s deconfined excitation. We then present a third example, the H2/Z4 model where we use an action of Z4 on H2 such that one of the fluxes become deconfined even in the presence of the link operator Cl.

The effect of confinement is due to the link operator Cl. A systematic way to deconfine all the fluxes is by working with Hamiltonians which do not have the Cl operators but just the vertex operator Av and the plaquette operator Bp. We then see that we have all the deconfined excitations of the quantum double models along with the vertex excitations due to the presence of the matter fields. These models will comprise the fourth set of examples. This is the simplest and most straightforward way to deconfine all the fluxes in contrast with the H2/Z4 case where a clever choice of input data and structure constant helped us deconfine the fluxes.

Finally we will show how to recover the quantum double Hamiltonians with only gauge fields by “switching off” the matter fields. These comprise the fifth set of examples. All the examples are constructed on the torus.

Before we write down the models we make a few comments about the ground state degeneracy of these models. The models we will construct are obtained from the transfer matrix which has the following form